﻿<p>The vertical parabolic segment is defined as a parabola using the inherited attributes from <i>IfcAlignment2DVerticalSegment</i> and the following additional curve parameters  <i>ParabolaConstant</i> as the minimum radius of the parabolic arc at its apex, and <i>IsConvex</i> to indicate the whether the parabolic arc defined a sag or a crest.</p>

<p>The parabolic arc is described by (see figure 1):</p>
<ul style="list-style-type: none">
 <li>R = signed <i>ParabolaConstant</i>, if <i>IsConvex</i> = true, then -R, if <i>IsConvex</i> = false, then +R as length measure</li>
 <li>s<sub>0</sub> = <i>StartDistAlong</i> as length measure</li>
 <li>z<sub>0</sub> = <i>StartHeight</i> as length measure</li>
 <li>g<sub>0</sub> = <i>StartGradient</i> as ratio measure<li>
</ul>

<p>The following equations are used for any point along the parabola:</p>
<ul style="list-style-type: none">
 <li>at any point s<sub>1</sub> along the <i>HorizontalLength</i>
 <li>the gradient g<sub>1</sub> = (s<sub>1</sub>-s<sub>0</sub>) / R + g<sub>0</sub></i>
 <li>the height z<sub>1</sub> = (s<sub>1</sub>-s<sub>0</sub>) * (g<sub>1</sub>+g<sub>0</sub>)/2 + z<sub>0</sub></i>
</ul>

<table>
<tr><td><img src="../../../figures/ifcalignment2dversegparabolicarc_fig1.png" /></td></tr>
<tr><td>
Legend:<br>
red: the parabolic curve with start point and calculated end point used as an vertical alignment</br>
yellow: the underlying unbounded parabola definition</br>
blue: the minimum circle defined by the parabola constant agreeing to the "is convex" fag, the constant, and the start gradient 
</td></tr>
<tr><td><p class="figure">Figure 4 &mdash; Alignment vertical segment parabola convex</p></td></tr>
</table>
<p>&nbsp;</p>
<table>
<tr><td><img src="../../../figures/ifcalignment2dversegparabolicarc-convex.png" /></td><td style="vertical-align: bottom">start point provided by <i>StartDistAlong</I> and <i>StartHeight</i><br>instanteneous gradient provided by <i>StartGradient</i>, <br>and length provided by <i>HorizontalLength</i></td></tr>
<tr><td><p class="figure">Figure 5 &mdash; Alignment vertical parabolic arc segment convex</p></td><td>&nbsp;</td></tr>
</table>
<p>&nbsp;</p>
<table>
<tr><td><img src="../../../figures/ifcalignment2dversegparabolicarc-concave.png" /></td><td style="vertical-align: bottom">start point provided by <i>StartDistAlong</I> and <i>StartHeight</i><br>instanteneous gradient provided by <i>StartGradient</i>, <br>and length provided by <i>HorizontalLength</i></td></tr></tr>
<tr><td><p class="figure">Figure 5 &mdash; Alignment vertical parabolic arc segment concave</p></td><td>&nbsp;</td></tr>
</table>